p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42⋊25D4, C23.509C24, C24.357C23, C22.2892+ 1+4, C22.2102- 1+4, C42⋊8C4⋊49C2, C23.8Q8⋊81C2, C23.Q8⋊35C2, C23.162(C4○D4), (C23×C4).413C22, (C22×C4).555C23, (C2×C42).596C22, C22.335(C22×D4), C23.23D4.44C2, C23.10D4.30C2, (C22×D4).186C22, C23.81C23⋊54C2, C24.C22⋊102C2, C2.78(C22.19C24), C23.63C23⋊110C2, C2.72(C22.45C24), C2.C42.238C22, C2.20(C22.31C24), C2.31(C22.49C24), C2.77(C22.46C24), (C2×C4).370(C2×D4), (C2×C42⋊C2)⋊34C2, (C2×C4).411(C4○D4), (C2×C4⋊C4).348C22, C22.385(C2×C4○D4), (C2×C22⋊C4).205C22, SmallGroup(128,1341)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42⋊25D4
G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, cac-1=a-1, dad=a-1b2, cbc-1=dbd=a2b-1, dcd=c-1 >
Subgroups: 500 in 270 conjugacy classes, 100 normal (22 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C24, C2.C42, C2.C42, C2×C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C23×C4, C22×D4, C42⋊8C4, C23.8Q8, C23.23D4, C23.63C23, C24.C22, C23.10D4, C23.Q8, C23.81C23, C2×C42⋊C2, C42⋊25D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C22×D4, C2×C4○D4, 2+ 1+4, 2- 1+4, C22.19C24, C22.31C24, C22.45C24, C22.46C24, C22.49C24, C42⋊25D4
►On 64 points
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)(25 26 27 28)(29 30 31 32)(33 34 35 36)(37 38 39 40)(41 42 43 44)(45 46 47 48)(49 50 51 52)(53 54 55 56)(57 58 59 60)(61 62 63 64)
(1 40 29 16)(2 37 30 13)(3 38 31 14)(4 39 32 15)(5 54 21 45)(6 55 22 46)(7 56 23 47)(8 53 24 48)(9 34 25 49)(10 35 26 50)(11 36 27 51)(12 33 28 52)(17 58 64 43)(18 59 61 44)(19 60 62 41)(20 57 63 42)
(1 47 28 63)(2 46 25 62)(3 45 26 61)(4 48 27 64)(5 33 44 16)(6 36 41 15)(7 35 42 14)(8 34 43 13)(9 19 30 55)(10 18 31 54)(11 17 32 53)(12 20 29 56)(21 52 59 40)(22 51 60 39)(23 50 57 38)(24 49 58 37)
(1 63)(2 19)(3 61)(4 17)(5 33)(6 51)(7 35)(8 49)(9 46)(10 54)(11 48)(12 56)(13 58)(14 42)(15 60)(16 44)(18 31)(20 29)(21 52)(22 36)(23 50)(24 34)(25 55)(26 45)(27 53)(28 47)(30 62)(32 64)(37 43)(38 57)(39 41)(40 59)
G:=sub<Sym(64)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64), (1,40,29,16)(2,37,30,13)(3,38,31,14)(4,39,32,15)(5,54,21,45)(6,55,22,46)(7,56,23,47)(8,53,24,48)(9,34,25,49)(10,35,26,50)(11,36,27,51)(12,33,28,52)(17,58,64,43)(18,59,61,44)(19,60,62,41)(20,57,63,42), (1,47,28,63)(2,46,25,62)(3,45,26,61)(4,48,27,64)(5,33,44,16)(6,36,41,15)(7,35,42,14)(8,34,43,13)(9,19,30,55)(10,18,31,54)(11,17,32,53)(12,20,29,56)(21,52,59,40)(22,51,60,39)(23,50,57,38)(24,49,58,37), (1,63)(2,19)(3,61)(4,17)(5,33)(6,51)(7,35)(8,49)(9,46)(10,54)(11,48)(12,56)(13,58)(14,42)(15,60)(16,44)(18,31)(20,29)(21,52)(22,36)(23,50)(24,34)(25,55)(26,45)(27,53)(28,47)(30,62)(32,64)(37,43)(38,57)(39,41)(40,59)>;
G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64), (1,40,29,16)(2,37,30,13)(3,38,31,14)(4,39,32,15)(5,54,21,45)(6,55,22,46)(7,56,23,47)(8,53,24,48)(9,34,25,49)(10,35,26,50)(11,36,27,51)(12,33,28,52)(17,58,64,43)(18,59,61,44)(19,60,62,41)(20,57,63,42), (1,47,28,63)(2,46,25,62)(3,45,26,61)(4,48,27,64)(5,33,44,16)(6,36,41,15)(7,35,42,14)(8,34,43,13)(9,19,30,55)(10,18,31,54)(11,17,32,53)(12,20,29,56)(21,52,59,40)(22,51,60,39)(23,50,57,38)(24,49,58,37), (1,63)(2,19)(3,61)(4,17)(5,33)(6,51)(7,35)(8,49)(9,46)(10,54)(11,48)(12,56)(13,58)(14,42)(15,60)(16,44)(18,31)(20,29)(21,52)(22,36)(23,50)(24,34)(25,55)(26,45)(27,53)(28,47)(30,62)(32,64)(37,43)(38,57)(39,41)(40,59) );
G=PermutationGroup([[(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24),(25,26,27,28),(29,30,31,32),(33,34,35,36),(37,38,39,40),(41,42,43,44),(45,46,47,48),(49,50,51,52),(53,54,55,56),(57,58,59,60),(61,62,63,64)], [(1,40,29,16),(2,37,30,13),(3,38,31,14),(4,39,32,15),(5,54,21,45),(6,55,22,46),(7,56,23,47),(8,53,24,48),(9,34,25,49),(10,35,26,50),(11,36,27,51),(12,33,28,52),(17,58,64,43),(18,59,61,44),(19,60,62,41),(20,57,63,42)], [(1,47,28,63),(2,46,25,62),(3,45,26,61),(4,48,27,64),(5,33,44,16),(6,36,41,15),(7,35,42,14),(8,34,43,13),(9,19,30,55),(10,18,31,54),(11,17,32,53),(12,20,29,56),(21,52,59,40),(22,51,60,39),(23,50,57,38),(24,49,58,37)], [(1,63),(2,19),(3,61),(4,17),(5,33),(6,51),(7,35),(8,49),(9,46),(10,54),(11,48),(12,56),(13,58),(14,42),(15,60),(16,44),(18,31),(20,29),(21,52),(22,36),(23,50),(24,34),(25,55),(26,45),(27,53),(28,47),(30,62),(32,64),(37,43),(38,57),(39,41),(40,59)]])
38 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | ··· | 4V | 4W | 4X | 4Y | 4Z |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | C4○D4 | C4○D4 | 2+ 1+4 | 2- 1+4 |
| kernel | C42⋊25D4 | C42⋊8C4 | C23.8Q8 | C23.23D4 | C23.63C23 | C24.C22 | C23.10D4 | C23.Q8 | C23.81C23 | C2×C42⋊C2 | C42 | C2×C4 | C23 | C22 | C22 |
| # reps | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 2 | 1 | 2 | 4 | 8 | 8 | 1 | 1 |
Matrix representation of C42⋊25D4 ►in GL6(𝔽5)
| 1 | 1 | 0 | 0 | 0 | 0 |
| 3 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 2 | 2 | 0 | 0 | 0 | 0 |
| 1 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 4 | 4 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 2 |
| 0 | 0 | 0 | 0 | 0 | 3 |
| 4 | 4 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 2 |
| 0 | 0 | 0 | 0 | 1 | 3 |
G:=sub<GL(6,GF(5))| [1,3,0,0,0,0,1,4,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[2,1,0,0,0,0,2,3,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[4,0,0,0,0,0,4,1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,2,0,0,0,0,0,2,3],[4,0,0,0,0,0,4,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,2,1,0,0,0,0,2,3] >;
C42⋊25D4 in GAP, Magma, Sage, TeX
C_4^2\rtimes_{25}D_4 % in TeX
G:=Group("C4^2:25D4"); // GroupNames label
G:=SmallGroup(128,1341);
// by ID
G=gap.SmallGroup(128,1341);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,253,568,758,723,675,248]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=d^2=1,a*b=b*a,c*a*c^-1=a^-1,d*a*d=a^-1*b^2,c*b*c^-1=d*b*d=a^2*b^-1,d*c*d=c^-1>;
// generators/relations